Proposal for a New Computer Olympiad Game –– Connect6 Proposed by I-Chen Wu in September 2005 Introduction and Rules Connec6, which can be viewed as a kind of 6-in-a-row game, was introduced by I-Chen Wu and Der-Yann Huang (Wu and Huang, 2005). The rules of Connect6, similar to those in Go-Moku, are very simple as follows. • Players and stones: There are two players. The first player, called Black here, holds a set of black stones, like Go or Go-Moku games. The second player, called White here, holds a set of white stones. • Game boards: In theory, the size of game board can be infinite. However, practically, it is hard to support such a game board. Therefore, our proposal is as follows. o For casual players, simply use 19×19 Go boards. o For professional players, use 59×59 boards. The reason for this size for professional players is in the following two senses. Players can tile up 9 Go boards to play, if games are played without the help of computers. In addition, judging from some current games, such as Go and Chess, we roughly estimate at most hundreds of moves per game in the extreme cases. Hundreds of moves inside a 59×59 board for Connect6 seem reasonable. o In order to make programmers easier to handle, 19×19 Go boards are initially recommended for Computer Olympiad game contests. 59×59 boards may be used in the future. • Game moves: Black plays first and places only one black stone on one unoccupied intersection, also called square in the rest of this document. Subsequently, Black and White alternately place two of their own stones on two unoccupied squares. • Game winning: The one who first gets six consecutive stones (horizontally, vertically or diagonally) of her/his own wins. Unlike Renju, a variant of Go-Moku for professionals, no extra rules are needed to be imposed on Connect6, since the rules of Connect6 are fair enough. Fairness We first review the fairness problem of Go-Moku and then discuss the fairness of Connect6. Fairness has been a major issue for Go-Moku, even though it has been a popular game. • In the free style of rules (without any restriction to Black), it has been well known that the game favors Black. Theoretically, it was proved (Allis, 1994; Allis, Herik, and Huntjens, 1996) that Black wins in the free style. • To reduce the unfairness problem, the Japanese professional Renju association added some rules (Japanese professional Renju association, 1903) to restrict the play of Black for professional players. For example, Black is prohibited to play double three and double four. In fact, Renju still favors Black, from the experiences of professionals. It was also proved (Wágner, Virág, 2001) that Black still wins. • The Renju International Federation (RIF) changed the rules (RIF, 1998) to make it fairer by imposing some opening rules for the first five moves. • In 2003, RIF requested a proposal (RIF, 2003) for better opening rules again and listed the requirements for new rules. This indicates that it is still hard to define a fair rule for the game. In fact, adding more rules makes it harder to learn the game. The fairness problem for Go-Moku or Renju also has a side effect. • Reduce the board size. It was argued (Sakata and Ikawa, 1981) that a larger board increases black’s advantage which resulted in the standard Renju board size of 15x15. However, a smaller board reduces the complexity of the game. Consequently, it becomes easier to solve the game. Herik, Uiterwijk, and Rijswijck gave a definition of fairness (Herik, Uiterwijk, and Rijswijck, 2002) as follows: A game is considered a fair game if it is a draw and both players have a roughly equal probability on making a mistake. From this, Connect6 is argued to be fair in the following two senses: • One player always has one more stone than the other after making each move. • The initial breakaway (means to play far away from the initial Black stone) does not apparently favor White. It was proved in (Wu and Huang, 2005) that if White makes an initial breakaway, Black wins. Note that if White did not have penalty by making an initial breakaway, the game would be just like a game that White places two stones initially. In practice, based on the current empirical experiences on Connect6 so far, it has not yet been identified that one player takes clear advantage over the other. Since the game was presented (Wu and Huang, 2005), tens of thousands of players, including many Renju dan players, have played Connect6 over an online game system (ThinkNewIdea Limited 2005). No players have yet been able to show winning sequences for Connect6. Complexity The state-space complexity of Connect6 is very high, since the board size can be very large. For casual players, the size is already 19×19, the same as the Go board. So, the state-space complexity is also 10172, the same as that in Go (Herik, Uiterwijk, and Rijswijck, 2002). For professional players, the size is 59×59, which makes the state-space complexity roughly as high as 101600. Now, investigate the game-tree complexity. Assume that the averaged game length is still 30, the same as the estimation for Go-Moku (Allis 1994). Then, the number of squares chosen to place one stone is about 300, and the number of choices of one move is about (300*300/2). Thus, the game-tree complexity is about (300*300/2)30 ~ 10140, much higher than 1070 for Go-Moku (Herik, Uiterwijk, and Rijswijck, 2002). Also, for the board used by professionals, this complexity is much higher. Example Games The following are played by two computers. White wins in the following case. Black wins in the following case. Following are some Tsumegos, won by White (Lee 2005). Current Status • The first Connect6 program was written (Wu and Huang 2004) in 2004. • The first scientific paper on Connect6 (Wu and Huang 2005) was published in ACG’11 in 2005. • The homepage for Connect6 is in www.connect6.org or connect6.csie.nctu.edu.tw. • The first online game server for Connect6 is www.cycgame.com/connect6 in 2005. Tens of thousands of players have played in this site. • The first Connect6 discussion group including Joseki/Tsumego was formed by Lee (Lee 2005). • The first Connect6 game contest is being proposed to the Computer Olympiad by this document. Expected Participants for Connect6 in the Computer Olympiad • I-C. Wu and D.-Y. Huang. • S.C. Hsu. • S. Lin. • S.J. Yen. References Books and Papers • Allis, L.V. (1994). Searching for Solutions in Games and Artificial Intelligence. Ph.D. Thesis, University of Limburg, Maastricht, The Netherlands. ISBN 9-0900-7488-0. • Allis, L.V., Herik, H.J van den, and Huntjens, M.P.H. (1993). Go-Moku and Threat-Space Search. Report CS 93-02, Department of Computer Science, Faculty of General Sciences, University of Limburg, Maastricht, The Netherlands. ISSN 0922-8721. • Allis, L.V., Herik, H.J. van den, and Huntjens, M.P.H. (1996). Go-Moku Solved by New Search Techniques. Computational Intelligence: An International Journal. Vol. 12, No. 1, pp. 7-24. Special Issue on Games. Blackwell Publishers. ISSN 0824-7935. • Allis, L.V., Meulen, M. van der, and Herik, H.J. van den (1994). Proof-Number Search. Artificial Intelligence, Vol. 66, No. 1, pp. 91-124. ISSN 0004-3702. • Herik, H.J. van den, Uiterwijk, J.W.H.M. and Van Rijswijck, J. (2002). Games solved: now and in the future. Artificial Intelligence, Vol. 134, pp. 277-311. • Japanese Professional Renju Association, (1903). History of Renju Rules, http://www.renjusha.net/database/oldrule.htm. • Lee, T.W. (2005) Joseki and Tsumegos for Connect6 (in Chinese). http://groups.msn.com/connect6/. • Nosovsky, A. (2002). Go-moku Still Alive. ICGA Journal, Vol. 25, No. 1, pp. 47-48. • Pluhar, A. (1994). Generalizations of the game k-in-a-row, Rutcor Res. Rep. 15-94. • Pluhar, A. (2002). The accelerated k-in-a-row game, Theoretical Computer Science. 271 (1-2) 865-875. • Renju International Federation (1998). The International Rules of Renju, http://www.renju.nu/rifrules.htm. • Renju International Federation (2003). Agenda for the RIF General Assembly, http://www.renju.nu/wc2003/MOM_RIF_030805.htm. • Sakata, G. and Ikawa, W. (1981). Five-in-a-Row, Renju. The Ishi Press, Inc. Tokyo, Japan. • ThinkNewIdea Limited (2005). CYC game web site (in Chinese). http://www.cycgame.com. • Wágner and Virág (2001). Solving Renju. ICGA Journal, Vol. 24, No. 1, pp. 30-34. • Wu, I-C., and Huang, D.-Y. (2005) A New Family of k-in-a-row Games. The 11th Advances in Computer Games (ACG11) Conference, Taipei, Taiwan. Web Sources • www.connect6.org or connect6.csie.nctu.edu.tw: A Connect6 web site. • www.cycgame.com/connect6: The first web site for Connect6 online games. .